Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,21)(16,26)(17,22)(18,27)(19,23)(20,24)(25,28);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,16,19,25,18,24)(14,20,27,28,23,26,22,21);(7,12)(9,10)(13,16,17,25,19,24,18,15)(14,21,27,20,23,28,22,26);(5,7)(6,10)(8,12)(9,11)(13,17,19,18)(14,22,23,27)(15,16,25,24)(20,21,26,28);(7,12)(9,10)Polycyclic presentation:
2^9|b^2=fh,c^2=gi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6514428);